Slender structures that are subject to mechanical excitations tend to exhibit transverse waves traveling along their lengths. A "slender" structure is one having a length that is relatively large, as compared to its size in any perpendicular dimension. The well known example of a guitar string experiences transverse waves. A guitar string is fixed at both ends. If the string is pulled from its rest position transverse to its length, and released, the string vibrates back and forth from one side of the rest position to the other. This is an example of lightly damped free vibration. For the guitar string such vibration occurs at particular frequencies, the natural frequencies in cycles per second or Hertz ("Hz"), which are given by the following equation: ##EQU1## where: T=tension (in units of force); .rho..sub.c =mass per unit length; L=length of string between fixed points; and n is an integer. The lowest natural frequency (n=1) is referred to as the "fundamental" frequency. The other natural frequencies are often referred to as "higher harmonic" frequencies. The sound heard upon plucking a guitar string includes the fundamental and several of the higher harmonic frequencies.
Strings under tension also support traveling waves, which one may generate and watch by periodically shaking one end of a long rope under low tension. In fact, the side-to-side motion of a guitar string can be represented mathematically as the sum of two waves of equal amplitude and frequency traveling in opposite directions. The sum of the two produces what is referred to as a "standing wave."
Standing waves are also produced when a traveling wave of a particular frequency and amplitude encounters a fixed end (a "termination"). There, the wave is reflected, producing a wave traveling in the opposite direction. The sum of the incident and reflected waves create a standing wave pattern, which is usually of greater amplitude than the incident wave and may lead to structural damage. One of the important features of the present invention is that it prevents waves from being reflected. A guitar string exhibits nearly perfect reflections at its terminations.
Inspection of equation (1) given above shows that for the natural frequencies, the length of the string is an integer multiple of half wave lengths. (A wave length .lambda. is the distance from the crest of one wave to the crest of the next, just as one sees on the surface of the ocean.) The fundamental natural frequency of a string is the frequency having a wavelength that is exactly equal to twice the length of the string.
The notion of natural frequencies is important in the context of this invention, because it is very easy to create large amplitude vibration of a tensioned string, cable, rope, ribbon or similar slender structure by providing an external periodic exciting force at one of the natural frequencies of the structure. This large amplitude response is due to an effect known as "dynamic amplification." One way to diminish or eliminate dynamic amplification is to reduce the efficiency of wave reflection at terminations. The present invention absorbs, rather than reflects the energy in waves which encounter the terminations or boundaries.
The guitar string has been used as an instructive example. The principle is also applicable to ship anchor lines, power transmission cables, and even drilling pipe suspended beneath a ship. In fact, it is applicable to any long slender structure capable of supporting traveling waves which cause movement perpendicular to the long axis of the structure, such as the lateral motion of a cable being towed through the water, or the vibration of an above-ground pipeline exposed to the wind.
It is well understood that for most mechanical structures, such as elevator cables, bridge suspension cables, high tension wires and nautical tow lines, it is undesirable to subject the structure to excitation forces that are near to a natural frequency. If that occurs, the structure may exceed its desired safe operating range. Sometimes, the vibration may lead to structural failure, with the structure vibrating free of its moorings, or experiencing fatigue failure. For this reason, engineers examine the frequencies of likely excitations, such as would be caused by traffic, normal use, earthquakes, wind, currents in water, ocean waves, etc., and try to design the structure and components thereof so that their natural frequencies differ greatly from any excitation frequencies that are likely to arise. This is not always possible and sometimes it is necessary to deal directly with the coincidence of exciting forces and natural frequencies. The problem is prevalent in connection with the task of towing objects through a fluid, such as in the water or air behind a ship or an airplane.
FIG. 1 shows schematically a ship 102 that is towing an object 104, such as a sonar sensor, magnetometer, or underwater camera, etc., in a direction indicated by an arrow V, at a velocity v through the water. The tow line or tow cable 106 is terminated at the boat at 108 and at the towed object 104 at 110, such that the ends of the tow line can not move with respect to either the ship or the towed object. In other words, its ends are fixed with respect to these objects, just like a guitar string, as discussed above. These objects are referred to below as "reference bodies." The ship is further classified as a stationary reference body, because it does not move in response to vibrations in the slender structure. Some towed objects are referred to as "movable reference bodies, because they are small enough so that they do move in response to vibrations in the slender structure. The principals of the invention apply to connections between slender structures and both types of reference bodies.
The tow line 106 experiences a tension T along its length, which is a function of the weight and buoyancy of the tow line and of the towed object 104, and the drag caused by the interaction of the tow line and the water through which it moves. The drag is affected by many things, including the diameter and surface characteristics of the tow line (i.e. smooth, fuzzy, ridged, etc.), the viscosity and density of the water and the velocity of the tow line through the water.
As a result of the flow of water around the tow line, eddies 112, also known as vortices, are shed in the wake of the tow line, i.e. behind it as it is pulled in direction V. The formation of the eddies causes transverse, periodic, lift forces on the tow line. If the eddies become correlated with each other with respect to time and their location along the length of the tow line, the resulting net force increases, leading to greater vibration. The frequency of the lift force, N.sub.s, may be predicted from the empirical relationship given below, ##EQU2## where: N.sub.s =the vortex shedding frequency in Hz; v=the velocity of the fluid perpendicular to the cable; D=the diameter of the cable; and S.sub.t =the Strouhal number, which is an empirical value that is known for most conditions. For circular cylinders S.sub.t is between 0.1 and 0.3 for most circumstances relevant here. See generally Blevins, Flow Induced Vibration, Van Nostrand Reinhold Co., 1977.
Large amplitude vibration results when the lift force frequency N.sub.s given above coincides with a natural frequency of the structure. This is because the standing wave motion of the structure at its natural frequency is able to synchronize the formation of the vortices and hence the lift forces along the length of the cylinder. This results in much larger motion than is possible with uncorrelated shedding of many vortex cells along the length. This phenomena is referred to as "lockin." However, it is not necessary for the vortex shedding frequency as given by the equation above to exactly equal a natural frequency. If the shedding frequency is within approximately +/-25% of a natural frequency, the shedding frequency may shift to the nearest natural frequency of the tow line, and the otherwise non-correlated lift and drag forces may become correlated along the entire length of the cable, resulting in substantial vibration.
Uncorrelated lift forces cause much less vibration than correlated lift forces resulting from lockin. Lockin may occur not only at the fundamental natural frequency of the tow line, but also at higher natural frequencies. Thus, as the fluid velocity changes, the lockin phenomena may jump from one natural frequency to the next. It is possible for a tow line to be continuously "locked in" to the eddy disturbances, regardless of the speed at which the tow line moves through the fluid, or the tension applied to the tow line. In cases like this it is usually impossible to reduce vibration by attempting to avoid natural frequencies.
Vibrations cause many problems. Principally, they add to the drag force between the fluid and the tow line, thereby requiring more power to attain the desired speed than would be necessary absent the vibrations. If the vibration becomes large enough, the tow line may fail due to abrasion or fatigue at its attachment points, with the ship or with the towed object, resulting in loss or damage of the towed object. Often, the device being towed is a sonar or other data gathering instrument. The vibrations produce noise that is received by the instrument, thereby obscuring the signal sought to be measured. The problem of flow-induced vibration is discussed in general in the text by Blevins, previously cited.
The present invention prevents the formation of standing waves, by absorbing rather than reflecting incident wave energy at the terminations of the towline, thus preventing lockin and reducing vibration. Many attempts have been made to overcome this nautical instance of the general problem which the invention addresses.
One general class of solutions tries to minimize the eddies that normally form, by attaching airfoil shaped fairings along the length of the tow line. The fairings, which are clipped onto the line, prevent the eddies from forming and thereby eliminate the periodic excitation. This approach has many drawbacks. Tow lines are normally stored on a roll, which is formed by winching a length of line around a drum. In use, lengths of tow line are spooled off from the drum, or winched back onto the drum as needed to change the length of the tow line. The fairings can not be attached to the tow line when it is winched up through the various guides and pulleys necessary to handle the lines. Further, they cannot remain attached to the line when it is rolled onto the drum. Thus, they must be attached as the tow line is let out, and removed as the tow line is reeled in. This presents a cumbersome and sometimes dangerous task. Further, the fairings must be very large if the tow line is of a large diameter. This presents significant handling, manufacturing, storage and safety complications. In applications relating to aircraft, the added weight of the fairings is also undesirable.
A second general class of solutions tries to reduce transverse vibrations by attaching long hair-like filaments to the entire surface of the tow line (so that it resembles a normal tow line covered with bristles or spaghetti). A principal problem with this general type of tow line is that it greatly increases the drag which must be overcome by the ship motor. It also makes winding the tow line on a drum and passing it through a pulley more difficult but sometimes not impossible. Addition of other structures, such as plastic pennants or flags has also been attempted. Another type of special tow line is made with helical strakes or ridges wrapped along the length of the tow line. The strakes tend to disorganize the forming eddies so that they are not correlated along the length of the tow line, thus reducing the periodic nature of the transverse forces caused by the eddies. A common embodiment of such strakes used to overcome a related problem can be seen spiraling up around smoke stacks, to minimize the effect of wind forces on the smoke stacks. In water such strakes reduce but generally do not sufficiently eliminate the problem.
A third class of solutions which is used in wind driven vibration problems, but not in water, is the "Stockbridge Damper." This is a device often seen hanging from the mid spans of high voltage power lines. Mechanically it is simply a mass suspended from the cable by a spring and damping element in parallel. The mass is free to move and acts as what is known as a dynamic absorber. See generally, J. P. Den Hartog, Mechanical Vibrations, p.93, (1956). The Stockbridge Damper absorbs energy from the waves in the cable in a band of frequencies near the natural frequency of the mass and spring system. It will not operate at a non-moving termination and it is not optimized so as to minimize the reflection coefficient. It does, under some circumstances, reduce vibration in wind driven cases. However, it is fundamentally different from the present invention.
The example of a tow line has been used here. The problem and typical solutions are applicable in many other structures including guy wires on towers and bridges, expanses of power wires, pipelines, etc. The problem is also present on oceanographic moorings, offshore oil drilling risers and marine pipelines.